On Autopolar Polyedra. [Abstract] [abstract]

Thomas P. Kirkman
1856 Proceedings of the Royal Society of London  
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more » ... out Early Journal Content at http://about.jstor.org/participate--jstor/individuals/early-journal--content. JSTOR is a digital library of academic journals, books, and primary source objects. JSTOR helps people discover, use, and build upon a wide range of content through a powerful research and teaching platform, and preserves this content for future generations. JSTOR is part of ITHAKA, a not--for--profit organization that also includes Ithaka S+R and Portico. For more information about JSTOR, please contact support@jstor.org. 217 2. That the lunar diurnal variation in each of the three elements constitutes a double progression in each lunar day; the declination having two easterly and two westerly maxima, and the inclination and total force each two maxima and two minima between two successive passages of the moon over the astronomical meridian; the variation passing in every case four times through zero in the lunar day. The approximate range of the lunar-diurnal variation at Toronto is 38" in the declination, 4"'5 in the inclination, and '000012 parts of the total force. 3. That the lunar-diurnal variation thus obtained appears to be consistent with the hypothesis that the moon's magnetism is, in great part at least if not wholly, derived by induction from the magnetism of the earth. That there is no appearance in the lunar-diurnal variation of the decennial period, which constitutes so marked a feature in the solar diurnal variations. XVII. "On Autopolar Polyedra." By the Rev. THOMAS P. KIRKMAN, M.A. Communicated by ARTHUR CAYLEY, Esq., F.R.S. Received June 19, 1856. (Abstract.) An autopolar polyedron is such, that any type or description that can be given of it remains unaltered, when summits are put for faces, and faces for summits. To every /-gon B in it corresponds a /3-ace b (or summit b of 3 edges), which may be called the pole of that XVII. "On Autopolar Polyedra." By the Rev. THOMAS P. KIRKMAN, M.A. Communicated by ARTHUR CAYLEY, Esq., F.R.S. Received June 19, 1856. (Abstract.) An autopolar polyedron is such, that any type or description that can be given of it remains unaltered, when summits are put for faces, and faces for summits. To every /-gon B in it corresponds a /3-ace b (or summit b of 3 edges), which may be called the pole of that ,3-gon; and to every edge AB, between the a-gon A and the 3-gon B, corresponds an edge ab, between the a-ace a and the /3-ace b. Two such edges are called a gamic pair, or pair of gamics. The enumeration of autopolar p-edra is here entered upon as a step towards the determination of the number of p-edra. The theorems following are established, and shown to be of importance for the solution of the general problem. THEOREM I.-No polyedron, not a pyramid, has every edge both in a triangle and in a triace.
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